What's the use of unit vectors?

• Odette
In summary, the conversation discusses the importance and use of unit vectors in a 3D vector equation. The first equation without unit vectors may not fully describe the vector, and unit vectors are implied in an orthonormal basis. Unit vectors are essential for specifying both the direction and magnitude of a vector.
Odette
Homework Statement
Hallo. Can somebody explain to me what's the importance-use of unit vector in the below (second) equation? Why isn't the first equation just enough to describe r? What's the reason for unit vector to even exist?
Relevant Equations
in the photos
Homework Statement: Hallo. Can somebody explain to me what's the importance-use of unit vector in the below (second) equation? Why isn't the first equation just enough to describe r? What's the reason for unit vector to even exist?
Homework Equations: in the photos

##x, y, z## are numbers. E.g. if ##x =1, y =2, z=3##, then ##r =6##. Which doesn't mean a lot in terms of 3D vectors.

##\vec{r} = (x, y, z)## is an alternative that doesn't use unit vectors explicitly, although they are implied in terms an orthonormal basis.

Suppose I want a constant vector of length a that points in the x directton?
$$\vec{a}=a\hat{x}$$
Or a variable vector of size y that points in z direction? $$\vec{b}=y\hat{z}$$
You really need all the symbols.

Every vector has both a direction and a magnitude. For example, the "x" is the scalar quantity that tells you the magnitude of the x component of r, and the x with the ^ tells you the scalar points in the x direction. So, for a vector r, it has a magnitude of "r" multiplied by the r with the ^ to tell you the direction "r" points in.

1. What is a unit vector?

A unit vector is a vector that has a magnitude of 1 and is used to indicate direction in a specific coordinate system. It is often denoted by adding a hat (^) symbol above the vector symbol.

2. How are unit vectors useful in physics?

In physics, unit vectors are used to represent the direction and orientation of physical quantities such as velocity, force, and acceleration. They allow us to break down a vector into its components and analyze its direction and magnitude.

3. Can unit vectors be used in multiple dimensions?

Yes, unit vectors can be used in multiple dimensions. In two-dimensional space, there are two unit vectors, i and j, which represent the x and y directions respectively. In three-dimensional space, there are three unit vectors, i, j, and k, which represent the x, y, and z directions respectively.

4. How do unit vectors relate to vector operations?

Unit vectors are often used in vector operations such as addition, subtraction, and multiplication. They can be used to break down a vector into its components, making it easier to perform these operations and understand the resulting vectors.

5. Are unit vectors unique?

Yes, unit vectors are unique as they have a magnitude of 1 and a specific direction. However, there can be multiple unit vectors in a given direction, as long as they have the same direction and magnitude.

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